61.3.17 problem Problem 18

Internal problem ID [15316]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.5 Laplace transform. Homogeneous equations. Problems page 357
Problem number : Problem 18
Date solved : Thursday, October 02, 2025 at 10:11:34 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y^{\prime \prime }+20 y^{\prime }+51 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-5 \\ \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 16

ode:=2*diff(diff(y(t),t),t)+20*diff(y(t),t)+51*y(t) = 0; 
ic:=[y(0) = 1, D(y)(0) = -5]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = {\mathrm e}^{-5 t} \cos \left (\frac {\sqrt {2}\, t}{2}\right ) \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 19

ode=2*D[y[t],{t,2}]+20*D[y[t],t]+51*y[t]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-5}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-5 t} \cos \left (\frac {t}{\sqrt {2}}\right ) \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(51*y(t) + 20*Derivative(y(t), t) + 2*Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -5} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = e^{- 5 t} \cos {\left (\frac {\sqrt {2} t}{2} \right )} \]